Numerical range decomposition via primitive ideals and irreducible representations

Introduction

One of the main tools of operator theory has been the numerical range of a bounded linear operator, which is an essentially single subset of the complex plane, carrying at the same time spectral, geometric and algebraic information about the operator [5,8] . In a Hilbert space, the classical numerical range of a quadratic form is the set of values that it assumes on the unit vectors and by the Toeplitz-Hausdorff theorem this set is always convex [9] . For inner productfree underlying space, Gunter Lumer's original idea was taken up by Frank F. Bonsall and John Duncan to develop the spatial numerical range, normed by norming pairs of vectors which was described in their two monographs entitled Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras [1] and [2] respectively and was documented in great detail. In this generality convexity is not automatic. On the Banach space of square-summable sequences, the spatial numerical range of the unilateral shift is the open unit disc but this set can be neither closed nor convex on a general Banach space [5] . Due to the possible failure of convexity, it is common to turn to the topological closure or the closed convex hull of the spatial numerical range. This closed convex hull is exactly that of the algebra numerical range [6] which is generated by taking the operator at all states of the enveloping operator algebra [1,2] and Bonsall and Duncan proved that the natural goal of any decomposition theory is to recover this convex envelope from smaller local pieces of the algebra [7] . One initial strategy is to divide by a closed two-sided ideal and to take the algebra range of the quotient. The quotient numerical range thus generated is always nonempty, compact and convex and for prime ideals it contains always the original spatial numerical range in the closed convex hull. The problem is that the zero ideal is prime so the entire family of prime ideals reduces to the closed convex hull without providing any finer structure and the set of prime ideals has no natural topology that links one quotient to another [3] . In recent years the interest in the numerical range and its convexity properties has not decreased but the questions raised by the numerical range remain open. In the non-unital setting, the convexity of the numerical range is sensitive whilst it is known that the numerical range of a convex Algebra is convex, the same is not known to be true of the spatial numerical range of a non-unital normed algebra. The question being open even for the Beurling algebra L¹ (G, ω) [13] . The question of when a numerical range provides a spectral set in the sense of von Neumann's inequality has also been extended from operators on Hilbert spaces to general Banach algebras with positive answers for matrix algebras and for the Calkin algebra and a family of counterexamples for other Banach algebras by Blazhko at al. [16] . A related issue is the instability of the behaviour of the spectrum when considered as a function of quotientconstruction. Brits constructed a general one demonstrating that for any given Banach algebra adding an appropriate inessential ideal in an algebra can make the spectrum function discontinuous [12] . On the representation-theoretic side; Farenick, Li, and Singla [14] have been actively developing boundary representations and matrix or operator-system ranges for finite-dimensional operator systems and the development of the extreme-point and boundary-representation machinery continues for which representations detect the boundary of a convex envelope as developed by Davidson and Kennedy [15] . Furthermore, the primitive ideal space and its Jacobson topology are also of current interest and recently Chen and Li have explicitly computed this space along with its Jacobson topology for the semigroup crossed product C*-algebra of an integral domain [17] . None of these recent papers however focuses on the primitive ideal space through the Jacobson topology to aid in the reconstruction of the numerical range itself and this is exactly what the present paper fills. The prime ideal framework retains only the closed convex hull of the spatial numerical range in a trivial manner and without any topological structure in the arrangement of the various quotients relative to each other. Restricting attention to primitive ideals both flaws are corrected simultaneously. The kernels of the algebraically irreducible representations are always prime and the quotienting by them controls the numerical range and the Jacobson (or hull-kernel) topology is a quasi-compact and T₀ topology on the totality of primitive ideals recording how one primitive ideal is contained in the closure of another [10,11] . In this paper, we then make a first attempt at numerical range decomposition in a primitive ideal setting. It is shown that the assignment of a primitive ideal to its associated quotient numerical range is order-reversing with respect to the specialization order on the primitive ideal space that the closed convex hull of the spatial numerical range is exactly recovered by the full primitive family and that the zero ideal being itself primitive is precisely why no finer reconstruction is possible from that family alone. That restricting to a separating subfamily of prime ideals one whose associated quotient-factoring states are weak*-dense in the state space and reconstructs the same closed convex hull while genuinely excluding the zero ideal thereby producing a nontrivial decomposition and that boundary representations realized through fibre numerical ranges attached to irreducible representations identify precisely the extreme points of the convex envelope. In each of these findings explicit corollaries and examples are provided and the paper ends with consolidating the main claims into one unifying theorem.

Notations, Definitions and Preliminaries

Throughout this paper, ℂ denotes the field of complex numbers and ℝ the field of real numbers. Let 𝑋 denote a complex Banach space with dim𝑋 ≥ 2, 𝑋 * its continuous dual space, and 𝐵(𝑋) the unital Banach algebra of all bounded linear operators on 𝑋 with identity operator 𝐼 𝑋 (or simply 𝐼) [4] . The standard dual pairing between 𝑥 ∈ 𝑋 and 𝑓 ∈ 𝑋 * is written as (𝑥, 𝑓) or 𝑓(𝑥). For any subset 𝑆 ⊆ ℂ or 𝑆 ⊆ 𝐵(𝑋), 𝑆 denotes its topological closure, 𝑐𝑜(𝑆) its convex hull, and 𝑐𝑜(𝑆) = 𝑐𝑜(𝑆) its closed convex hull. The boundary of 𝑆 ⊆ ℂ is denoted ∂𝑆, and 𝐸𝑥𝑡(𝐾) denotes the set of extreme points of a compact convex set 𝐾 ⊂ ℂ.The spectrum of an operator 𝑇 ∈ 𝐵(𝑋) is denoted by 𝜎(𝑇). The global numerical radius of an operator 𝑇 ∈ 𝐵(𝑋) is defined as 𝑣(𝑇) = sup{|𝜆|: 𝜆 ∈ 𝑊(𝑇)}. The algebra numerical range of 𝑇 in a unital Banach algebra 𝐴 is 𝑉(𝑇, 𝐴) with state space 𝑆(𝐴), while 𝑊(𝑇) and 𝑊 (𝑇) ̅̅̅̅̅̅̅̅ denote the spatial numerical range of 𝑇 and its topological closure, respectively. For a closed two-sided ideal 𝐼 ⊆ 𝐵(𝑋), [𝑇] 𝐼 = 𝑇 + 𝐼 denotes the coset of 𝑇 in the quotient Banach algebra 𝐵(𝑋)/𝐼, and 𝑊[𝑇] 𝐼 is the corresponding quotient numerical range. 𝐶𝑙𝑃𝑟𝑖𝑚𝑒(𝐴) and 𝑃𝑟𝑖𝑚(𝐴) denote the spaces of closed prime ideals and primitive ideals of 𝐴, respectively. For 𝑃 ∈ 𝑃𝑟𝑖𝑚(𝐵(𝑋)) with algebraically irreducible representation 𝜋 𝑃: 𝐵(𝑋) → 𝐵(𝑌 𝑃 ), 𝑉 𝑃 (𝑇) = 𝑊(𝜋 𝑃 (𝑇)) denotes the fibre numerical range.

Definition 2.1 (Spatial and Algebra Numerical Range; Bonsall & Duncan 1971). Let 𝑇 ∈ 𝐵(𝑋).

The spatial numerical range of 𝑇 is

The set of all states is denoted 𝑆(𝐵(𝑋)).

The algebra numerical range of 𝑇 ∈ 𝐵(𝑋) is Since𝐸 𝑥,𝑓 (𝐼) = 1, we have∥ 𝐸 𝑥,𝑓 ∥= 1, which implies𝐸 𝑥 ,𝑓 ∈ 𝑆(𝐵(𝑋)). Thus𝜆 = 𝐸 𝑥,𝑓 (𝑇) ∈ 𝑉(𝑇, 𝐵(𝑋)). This shows𝑊(𝑇) ⊆ 𝑉(𝑇, 𝐵(𝑋)). Since 𝑉(𝑇, 𝐵(𝑋)) is closed and convex by Lemma 2.2, taking the closed convex hull gives𝑐𝑜 𝑊(𝑇) ⊆ 𝑉(𝑇, 𝐵(𝑋)).

Step 2: Inclusion𝑉(𝑇, 𝐵(𝑋)) ⊆ 𝑐𝑜 𝑊(𝑇). Both 𝑉(𝑇, 𝐵(𝑋)) and 𝑐𝑜 𝑊(𝑇) are compact convex subsets ofℂ ≅ ℝ 2 . By the Hahn-Banach separation theorem, a compact convex set in ℝ 2 is uniquely determined by its support functions. For any𝜃 ∈ [0,2𝜋), define the support functionℎ 𝐴 (𝜃) = sup{𝑅𝑒(𝑒 -𝑖𝜃 𝑧): 𝑧 ∈ 𝐴}. By standard numerical radius estimations (cf. [1] , Theorem 9. For any two ideals 𝐼, 𝐽, ℎ𝑢𝑙𝑙(𝐼) ∪ ℎ𝑢𝑙𝑙(𝐽) = ℎ𝑢𝑙𝑙(𝐼𝐽). Proof. By Theorem 2.6, 𝑊[𝑇] 𝑃 ⊆ 𝑐𝑜 𝑊(𝑇) for every 𝑃 ∈ 𝑃𝑟𝑖𝑚(𝐵(𝑋)). Taking the union over all 𝑃 and then the closed convex hull yields:

For the reverse inclusion, Corollary 2.13 establishes that 0 ∈ 𝑃𝑟𝑖𝑚(𝐵(𝑋)). Setting 𝑃 = 0, the quotient algebra 𝐵(𝑋)/0 is canonically isometric to 𝐵(𝑋), so 𝑊[𝑇] 0 = 𝑉(𝑇, 𝐵(𝑋)). By Theorem 2.3, 𝑉(𝑇, 𝐵(𝑋)) = 𝑐𝑜 𝑊(𝑇). Thus:

Combining both inclusions (1) and (2) proves equality. □ Corollary 3.5 (Spectral Formula). For every 𝑇 ∈ 𝐵(𝑋), ⋃ 𝑃∈𝑃𝑟𝑖𝑚(𝐵(𝑋))

𝜎([𝑇] 𝑃 ) = 𝜎(𝑇).

Proof [18] , the extreme points of the closed convex hull of a compact set must be contained within the set itself. 𝑉 𝑃 (𝑇) = 𝑐𝑜 𝑊(𝑇).

Proof. For 𝑃 = 0 ∈ 𝑃𝑟𝑖𝑚(𝐵(𝑋)) (Corollary 2.13), 𝜋 0 = 𝑖𝑑 𝐵(𝑋) , so 𝑉 0 (𝑇) = 𝑊(𝑇). Thus: 𝑣(𝜋 𝑃 (𝑇)).

Proof. For any bounded set𝑆 ⊂ ℂ, sup 𝑧∈𝑆 |𝑧| = sup 𝑧∈𝑐𝑜(𝑆) |𝑧|. Applying this identity to Theorem 3.14 gives𝑣(𝑇) = sup 𝑃 𝑣(𝜋 𝑃 (𝑇)). □

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