A Characterization of Essential Matrix Ranges
Francis J. Narcowich, J. D. Ward · Bulletin of the London Mathematical Society · 1982
The essential numerical range [5, Section 34] of an operator on a Hilbert space can be viewed in two ways: first, as the numerical range of the operator's Calkin element; and second, as the intersection of the (algebra) numerical ranges for all operators which differ from the original by a compact one.Stampfli and Williams [11] originally defined the essential numerical range in the first way, and then showed that the second was equivalent to it [11, Theorem 9].In many instances the second characterization has been the more useful of the two and has been used by some authors (for example Lancaster [8]) even to define the essential numerical range.For an element of a C*-algebra, Bunce and Salinas [6] defined the essential matrix ranges (relative to an ideal) in much the same way as Stampfli and Williams defined the essential numerical range-in terms of the corresponding quotient-algebra matrix ranges.In this paper, we will show that each essential matrix range is an intersection of matrix ranges, an intersection analogous to that used in the second characterization of essential numerical ranges.To precisely formulate this characterization of an essential matrix range requires some notation and several definitions: Let n be a fixed, positive integer.Recall that the n-th matrix range [3, p. 301] for an element of a unital C*-algebra is the set of that element's images under all possible unital completely positive maps which take the C*-algebra into M n , the set of n x n complex matrices.If si is a unital C*-algebra, and if T e si, denote the n-th matrix range of T by W n (T).If,/ is a closed two-sided ideal in si, define the n-th essential matrix range of 7' to be the n-th matrix range for the equivalence class of T in si//; denote this by W e n {T) (see [6]).In addition, let / and /" denote the identities in si and M n , respectively; let ||-|| and ||-|| e denote, respectively, the norms in si; no distinction will be made between these two spaces.The following characterization of essential matrix ranges is the main result of this paper.THEOREM.Let si be a unital C*-algebra and let / be a closed two-sided ideal in si.For every T in si, W e n (T)= H W n (T + J).The proof, which will be given later, involves both a characterization of matrix ranges, via a generalized Lumer's formula, and the concept of a quasicentral approximate unit for an ideal.Quasicentral approximate units will be discussed next.Let {A\,} and {y v } be nets in a C*-algebra.Define {A',.} and {Y x ) to be equivalent if lim||X v -y^.| | = 0; denote equivalence by writing {A",.}