THE ALGEBRAIC STRUCTURE OF THE TENSOR’S PRODUCE ROOM AND THE MATRIX CHAMBER OF THE OPERATOR’S ROOM ARE LIMITED TO THE HILBERT SPACE

ALGEBRA STRUCTURE
FROM TENSOR’S RESULTS ROOM
AND THE MATRIX SPACE OF THE OPERATOR’S ROOM IS LIMITED
IN THE HILBERT ROOM

Sadiq Khan (033019)
Mathematics Study Program
FPMIPA
UPI Bandung

Abstract. Suppose H is Hilbert’s room, a Hilbert n-tuple space (Hn)is Hilbert’s space as well. By identifying each matrix in Mn(B(H)) as a limited operator in Hn it can be indicated that Mn(B(H)) is a subaljabar* of B(Hn). Furthermore, it can be shown that there is a canonical isomorphism* of Mn(B(H)) in B(Hn) which causes the structure of the subaljabar* Mn(B(H)) to be similar to the structure of the operator algebra B(Hn). Through canonical isomorphism* we define norm in Mn(B(H)) in such a way as the norm in B(Hn). With the same algebraic and norm structure, Mn(B(H)) can be seen as algebra C* as well as B(Hn). This problem will be even more interesting if Mn(B(H)) is seen as the resulting space tensor square matrix space of the complex space with algebra operator B(H) (notified space Mm,n ⊗ B(H)), then hilbert n-tuple space (Hn)is seen as the result of tensor of n-tuple complex space with Hilbert H space (notified Cn ⊗ H).

Keywords: Algebra-C*, Algebra Operator, Hilbert n-tuple Room, Limited Operator, Canonical Isomorphism*, Subaljabar*, Tensor Multiplication Room.

1. Introduction
Various approaches can be used to study operator algebra, among them through the approach of algebraic and analytical properties of the matrix over linear space. A n×n matrix over linear space R can be seen as a linear transformation from an n-tuple linear space into the same n-tuple linear space, thus the n× n matrix is the linear operator of the n-tuple vector space. Furthermore a n×n matrix over the limited operator space B(H) can be viewed as the operator in the Hilbert n-tuple space.

An interesting problem to discuss when the matrix is seen as a linear combination of tensor element i i b =Σα ⊗b, with i b ∈ B(H), and i α scalar matrix n × n element from Mn (matrix space over complex space). So if the matrix space is treated as the operator’s room in hilbert space, then the result may also be treated the same, as if the matrix space is an algebra-C*.

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