ListBox sortieren C-Sharp WPF

<Window x:Class="ListFilesWPF.MainWindow"  
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"  
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"  
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"  
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"  
        xmlns:local="clr-namespace:ListFilesWPF"  
        mc:Ignorable="d"  
        Title="MainWindow" Height="350" Width="525" Loaded="Window_Loaded">  
    <Grid>
        <ListView x:Name="lv" Margin="10">  
            <ListView.View>
                <GridView>
                    <GridViewColumn DisplayMemberBinding="{Binding Path=Name}" Width="200">  
                        <GridViewColumnHeader Click="GridViewColumnHeader_Click">Name</GridViewColumnHeader>  
                    </GridViewColumn>
                    <GridViewColumn DisplayMemberBinding="{Binding Path=LastWriteTime,StringFormat='dd.MM.yyyy HH:mm:ss'}" Width="150">  
                        <GridViewColumnHeader Click="GridViewColumnHeader_Click">LasttWriteTime</GridViewColumnHeader>  
                    </GridViewColumn>
                    <GridViewColumn DisplayMemberBinding="{Binding Path=CreationTime,StringFormat='dd.MM.yyyy HH:mm:ss'}" Width="150">  
                        <GridViewColumnHeader Click="GridViewColumnHeader_Click">CreationTime</GridViewColumnHeader>  
                    </GridViewColumn>
                </GridView>
            </ListView.View>
        </ListView>
    </Grid>
</Window>

using System;
using System.Collections.Generic;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.IO;
using System.ComponentModel;

namespace ListFilesWPF {
    /// <summary>
    /// Interaktionslogik für MainWindow.xaml
    /// </summary>
    public partial class MainWindow : Window {
        GridViewColumnHeader _lastHeader = null;
        ListSortDirection _lastDir = ListSortDirection.Ascending;
        private FileInfo[] files;

        public MainWindow() {
            InitializeComponent();
        }
        
        private void Window_Loaded(object sender, RoutedEventArgs e) {
            files = new DirectoryInfo(@"D:\MyFolder").GetFiles("*.pdf");  
            lv.ItemsSource = files;
        }
        
        private void GridViewColumnHeader_Click(object sender, RoutedEventArgs e) {
            var headerClicked = e.OriginalSource as GridViewColumnHeader;
            var columnBinding = headerClicked.Column.DisplayMemberBinding as Binding;
            _lastDir = (_lastDir == ListSortDirection.Ascending) ? ListSortDirection.Descending : ListSortDirection.Ascending;
            if(_lastHeader != headerClicked) {
                _lastDir = ListSortDirection.Ascending;
            }
            CollectionView view = (CollectionView)CollectionViewSource.GetDefaultView(lv.ItemsSource);
            view.SortDescriptions.Clear();
            view.SortDescriptions.Add(new SortDescription(columnBinding.Path.Path, _lastDir));
            _lastHeader = headerClicked;
        }
    }
}